# Strong convergence theorems for relatively nonexpansive mappings and Lipschitz-continuous monotone mappings in Banach spaces

- 12 Downloads

## Abstract

In this paper, we introduce an iterative process for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of the variational inequality for a Lipschitz-continuous, monotone mapping in a Banach space. We obtain a strong convergence theorem for three sequences generated by this process. Our results improve and extend the corresponding results announced by many others. A simple numerical example is given to support our theoretical results.

## Key words

Relatively nonexpansive mapping generalized projection monotone mapping variational inequality 2-uniformly convex## Preview

Unable to display preview. Download preview PDF.

## Notes

### Acknowledgement

This work was supported by the National Natural Science Foundation of China(11401157) and the Key Laboratory of Machine Learning and Computational Intelligence of Hebei Province in College of Mathematics and Information Science of Hebei University.

## References

- 1.Y. I. Alber and S. Reich, An iterative method for solving a class of nonlinear operator equations in Banach spaces,
*Panamer Math. J.*,**4**(1994), 39–54.MathSciNetzbMATHGoogle Scholar - 2.K. Ball, E. A. Carlen, and E. H. Lieb, Sharp uniform convexity and smoothness inequalities for trace norms,
*Invent. Math.*,**115**(1994), 463–482.MathSciNetCrossRefGoogle Scholar - 3.N. Buong, Strong convergence theorem of an iterative method for variational inequalities and fixed point problems in Hilbert spaces,
*Appl. Math. Comput.*,**217**(2010), 322–329.MathSciNetzbMATHGoogle Scholar - 4.L. C. Ceng, N. Hadjisavvas, and N. C. Wong, Strong convergence theorem by hybrid extragradient-like approximation method for variational inequalities and fixed point problems,
*J. Glob. Optim.*,**46**(2010), 635–646.MathSciNetCrossRefGoogle Scholar - 5.Y. Censor, A. Gibali, and S. Reich, Algorithms for the split variational inequality problem,
*Numer. Algorithms*,**59**(2012), 301–323.MathSciNetCrossRefGoogle Scholar - 6.Y. Censor, A. Gibali, and S. Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space,
*Optim. Methods Softw.*,**26**(2011), 827–845.MathSciNetCrossRefGoogle Scholar - 7.Y. Censor, A. Gibali, and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert spaces,
*J. Optim. Theory Appl.*,**148**(2011), 318–335.MathSciNetCrossRefGoogle Scholar - 8.J. M. Chen, L. J. Zhang, and T. G. Fan, Viscosity approximation methods for nonexpansive mappings and monotone mappings,
*J. Math. Anal. Appl.*,**334**(2007), 1450–1461.MathSciNetCrossRefGoogle Scholar - 9.C. J. Fang, Y. Wang, and S. Yang, Two algorithms for solving single-valued variational inequalities and fixed point problems,
*J. Fixed Point Theory Appl.*,**18**(2016), 27–43.MathSciNetCrossRefGoogle Scholar - 10.H. Iiduka, Acceleration method for convex optimization over the fixed point set of a nonexpansive mapping,
*Math. Program. Ser. A*,**149**(2015), 131–165.MathSciNetCrossRefGoogle Scholar - 11.H. Iiduka, A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping,
*Optimization*,**59**(2010), 873–885.MathSciNetCrossRefGoogle Scholar - 12.H. Iiduka and I. Yamada, A subgradient-type method for the equilibrium problem over the fixed point set and its applications,
*Optimization*,**58**(2009), 251–261.MathSciNetCrossRefGoogle Scholar - 13.H. Iiduka and I. Yamada, A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping,
*SIAM J. Optim.*,**19**(2009), 1881–1893.MathSciNetCrossRefGoogle Scholar - 14.H. Iiduka and W. Takahashi, Strong convergence studied by a hybrid type method for monotone operators in a Banach space,
*Nonlinear Anal-theor.*,**68**(2008), 3679–3688.MathSciNetCrossRefGoogle Scholar - 15.H. Iiduka and W. Takahashi, Strong convergence theorems for nonexpansive mappings and inversestrongly monotone mappings,
*Nonlinear Anal-theor.*,**61**(2005), 341–350.CrossRefGoogle Scholar - 16.H. Iiduka and W. Takahashi, Strong convergence theorems for nonexpansive nonself mappings and inverse-strongly monotone mappings,
*J. Conv. Anal.*,**11**(2004), 69–79.MathSciNetzbMATHGoogle Scholar - 17.H. Iiduka and W. Takahashi, Weak convergence of a projection algorithm for variational inequalities in a Banach space,
*J. Math. Anal. Appl.*,**339**(2008), 668–679.MathSciNetCrossRefGoogle Scholar - 18.G. M. Korpelevič, An extragradient method for finding saddle points and for other problems,
*Èkon, Mat. Metody*,**12**(1976), 747–756, (In Russian).MathSciNetGoogle Scholar - 19.R. Kraikaew and S. Saejung, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces,
*J. Optim. Theory Appl.*,**163**(2014), 399–412.MathSciNetCrossRefGoogle Scholar - 20.J. L. Lions and G. Stampacchia, Variational inequalities,
*Comm. Pure Appl. Math.*,**20**(1967), 493–517.MathSciNetCrossRefGoogle Scholar - 21.Y. Liu, Strong convergence theorem for relatively nonexpansive mapping and inverse-strongly-monotone mapping in a Banach space,
*Appl. Math. Mech-Engl.*,**30**(2009), 925–932.MathSciNetCrossRefGoogle Scholar - 22.P. E. Maing, A hybrid extragradient-viscosity method for monotone operators and fixed point problems,
*SIAM J. Control Optim.*,**47**(2008), 1499–1515.MathSciNetCrossRefGoogle Scholar - 23.S. Matsushita and W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space,
*J. Approx. Theory*,**134**(2005), 257–266.MathSciNetCrossRefGoogle Scholar - 24.N. Nadezhkina and W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz continuous monotone mappings,
*SIAM J. Optim.*,**16**(2006), 1230–1241.MathSciNetCrossRefGoogle Scholar - 25.K. Nakajo, Strong convergence for gradient projection method and relatively nonexpansive mappings in Banach spaces,
*Appl. Math. Comput.*,**271**(2015), 251–258.MathSciNetzbMATHGoogle Scholar - 26.Y. Takahashi, K. Hashimoto, and M. Kato, On sharp uniform convexity, smoothness, and strong type, cotype inequalities,
*J. Nonlinear Convex Anal.*,**3**(2002), 267–281.MathSciNetzbMATHGoogle Scholar - 27.W. Takahashi and M. Toyoda, Weak convergence theorem for nonexpansive mappings and monotone mappings,
*J. Optim. Theory Appl.*,**118**(2003), 417–428.MathSciNetCrossRefGoogle Scholar - 28.A. R. Tufa, and H. Zegeye, An algorithm for finding a common point of the solutions of fixed point and variational inequality problems in Banach spaces,
*Arab. J. Math.*,**4**(2015), 199–213.MathSciNetCrossRefGoogle Scholar